438 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



ber n in the two preceding formulae, let there be formed two series of equations, 

 as follows : 



^X„-2a;;2X, + ^X, = ^ Y„-2a;^ Y,.-|-;s Y^ = 



2^Y-2a;^'Y +;s^Y=0 



«'Y -2a:;^% + ^^Y=0 



Put 



^^X, -2 3:^^X2 + ^^X3=0 

 VX^-2a:«% + «%=0 



&c. &c. 



P = ^X, + ^^X^ + ^X3 ...... +^"-'X_,+^''X^ + , &c. 



Q=^Y, + ^Y^ + ^% +^"-'Y„_, + ^"Y„ + ,&c. 



Then, by adding into one sum each of the two series of equations, we have 



^(Yo+Q)-2a;Q + ^-Y, = 0. 

 From these two equations we find 



X 3 — Xo«" 



Now, Xo=l, X =«, 



Therefore, the values of P and Q are 



Q 



Y„=0, Y.=y 



P = 



{x — z) z 



l-2zz + z' 



Q 



y z 



l-2zz + z' 



Let the expansion of the fraction |— -^ ^ be 



C+C.;^ + C;^^ 



+c„_, ^" 



' + C z" 



n — ] 



+ C 2" + &c. 



we have now 



But 

 and 



P = a;Co^ + (a;C,-C„)^^ 

 Q=_^ Coz+y C^z 



P:=X| 2 + X2 Z' 



Q = Y,z +Y^z' 



Hence it follows that 



+ (a:C„_,-C„_,)^" + &c. 



+ X„ z" + &c. 

 + Y„ 2:" + &c. 



Y„=yC„_, L. 



To simplify, let us put u instead of 2 a? in the expression l — 2zz + z\ so that 

 it becomes l — (u—z)z, we have now by division = — ^ ;== — ^ equal to 



^ ^ ^ l — 2xz + z^ l — (u — z)z ^ 



l+^(^U-z)+z'(u-zy . . . +2r"-'(M-«)"-^ + ^'-'(M-;^)"-'+&C. 



Let A^and B„ be the coefficients of 2;" in the expansions of («— s)"~' and (m-^)""', 

 and let A, and B, be the coefficients of 2; in the expansions of («-«)""" and («—«)""', 



